Show a group is solvable iff quotient by its center is solvable

Question

Let $G$ be a group. Prove that

$G$ is solvable $\iff$ $G'$ is solvable $\iff$ $G/Z(G)$ is solvable.

$G'$ is the commutator subgroup.

I've figured out the first of the stated equivalences but don't really know where to start on the second.

Answer 1

If $G$ is solvable, so is every quotient - see this question: Solvable implies quotient group is solvable: Proof check. For the converse see this duplicate, with $H=Z(G)$. Or see this one:

For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?